Static image principle for anisotropic conducting half‐space problems: Impedance boundary

Abstract

The image principle developed for static problems involving an anisotropic half‐space and bounded by either a perfect electric or magnetic conductor is extended to problems with an anisotropic surface impedance boundary. Such a boundary can be applied to approximate a thin layer of anisotropic conducting material above the anisotropic half‐space. The problem is limited by requiring similar anisotropy for the surface impedance and the transverse part of the resistivity dyadic of the half‐space. It is seen that, instead of a point image for a point source, the image consists of a combination of a point image and a line image obeying an exponential law. The effect of the impedance surface on the potential field of a point source is considered in terms of a numerical example.

Electrostatic image theory for a sphere with impedance surface

“…To check the theory, let us take the limiting case a --f bo, whence the image theory given in [6] for a planar surface should emerge. In fact, writing…”

Static image method for layered isotropic and BI‐isotropic cylinders

“…In fact, applying the transformation, we have a correspondence between the two problems. So it suffices to make a transformation to the image which has been obtained before [7]. Let us denote the outer radius of the coated cylinder by a and the thickness by t .…”

Electrostatic Image Theory for an Anisotropic Boundary of an Anisotropic Half-Space

“…‡ The referenced article, [3], seems to be inaccurate at a few key points. The following list proposes some amendments.…”

“…This equals the result obtained in [3], where an affine transformation was used to solve the problem ‡ . The reference deals with static currents, but the result is applicable to electrostatics just by changing the symbols: Q for I 0 ,¯ r / To see that (70) is reducible to the exponential line (81), we repeat the limiting process ζ d → 0 for the complete solution.…”

“…Over the years limited special solutions have been worked out, for example, for an anisotropic halfspace over a perfectly electrically or magnetically conducting surface [1], for an isotropic half-space bounded by an anisotropic surface [2], and for a similarly anisotropic half-space and boundary [3]. Our proposed general solution, in contrast, will work with all physically justified medium parameters, and the known cases are built in the new theory, as will be shown.…”